This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 15460

2023 Dutch IMO TST, 4
Tags: number theory
Find all positive integers $n$, such that $\sigma(n) =\tau(n) \lceil {\sqrt{n}} \rceil$.

2015 India IMO Training Camp, 1
Tags: number theory , frobenius , additive combinatorics , additive number theory , additive representation , induction
Let $n \ge 2$ be an integer, and let $A_n$ be the set \[A_n = \{2^n - 2^k\mid k \in \mathbb{Z},\, 0 \le k < n\}.\] Determine the largest positive integer that cannot be written as the sum of one or more (not necessarily distinct) elements of $A_n$ . [i]Proposed by Serbia[/i]

2004 Regional Olympiad - Republic of Srpska, 3
Tags: number theory , combinatorics
An $8\times8$ chessboard is completely tiled by $2\times1$ dominoes. Prove that we can place positive integers in all cells of the table in such a way that the sums of numbers in every domino are equal and the numbers placed in two adjacent cells are coprime if and only if they belong to the same domino. (Two cells are called adjacent if they have a common side.) Well this can belong to number theory as well...

2023 Thailand October Camp, 2
Tags: number theory
A number is called [i]Norwegian[/i] if it has three distinct positive divisors whose sum is equal to $2022$. Determine the smallest Norwegian number. (Note: The total number of positive divisors of a Norwegian number is allowed to be larger than $3$.)

2001 Finnish National High School Mathematics Competition, 5
Tags: number theory
Determine $n \in \Bbb{N}$ such that $n^2 + 2$ divides $2 + 2001n.$

1999 BAMO, 1
Tags: number theory
Prove that among any $12$ consecutive positive integers there is at least one which is smaller than the sum of its proper divisors. (The proper divisors of a positive integer n are all positive integers other than $1$ and $n$ which divide $n$. For example, the proper divisors of $14$ are $2$ and $7$.)

1995 May Olympiad, 1
Tags: number theory , game , game strategy , divisor
Veronica, Ana and Gabriela are forming a round and have fun with the following game. One of them chooses a number and says out loud, the one to its left divides it by its largest prime divisor and says the result out loud and so on. The one who says the number out loud $1$ wins , at which point the game ends. Ana chose a number greater than $50$ and less than $100$ and won. Veronica chose the number following the one chosen by Ana and also won. Determine all the numbers that could have been chosen by Ana.

2010 Brazil National Olympiad, 2
Tags: polynomial , ceiling function , number theory , prime number , algebra
Let $P(x)$ be a polynomial with real coefficients. Prove that there exist positive integers $n$ and $k$ such that $k$ has $n$ digits and more than $P(n)$ positive divisors.

2016 Costa Rica - Final Round, N2
Tags: prime , number theory
Determine all positive integers $a$ and $b$ for which $a^4 + 4b^4$ be a prime number.

2004 Estonia National Olympiad, 1
Tags: gcd , lcm , number theory
Find all triples of positive integers $(x, y, z)$ satisfying $x < y < z$, $gcd(x, y) = 6, gcd(y, z) = 10, gcd(z, x) = 8$ and $lcm(x, y,z) = 2400$.

2014 JBMO Shortlist, 4
Tags: number theory , prime number
Prove that there are not intgers $a$ and $b$ with conditions, i) $16a-9b$ is a prime number. ii) $ab$ is a perfect square. iii) $a+b$ is also perfect square.

1962 All-Soviet Union Olympiad, 12
Tags: number theory , algebra , divisibility
Given unequal integers $x, y, z$ prove that $(x-y)^5 + (y-z)^5 + (z-x)^5$ is divisible by $5(x-y)(y- z)(z-x)$.

2023 Brazil Cono Sur TST, 1
Tags: number theory
Let $n = p_1p_2 \dots p_k$ be the product of distinct primes $p_1, p_2, \dots , p_k$, with $k > 1$. Find all $n$ such that $n$ is multiple of $p_1 - 1, p_2 - 1, \dots , p_k - 1$.

1997 Akdeniz University MO, 3
Tags: equation , number theory , sequence
$(x_n)$ be a sequence with $x_1=0$, $$x_{n+1}=5x_n + \sqrt{24x_n^2+1}$$. Prove that for $k \geq 2$ $x_k$ is a natural number.

1992 Miklós Schweitzer, 5
Tags: number theory , real analysis
Prove that if the $a_i$'s are different natural numbers, then $\sum_ {j = 1}^n a_j ^ 2 \prod_{k \neq j} \frac{a_j + a_k}{a_j-a_k}$ is a square number.

1988 Mexico National Olympiad, 5
Tags: number theory , coprime , greatest common divisor
If $a$ and $b$ are coprime positive integers and $n$ an integer, prove that the greatest common divisor of $a^2+b^2-nab$ and $a+b$ divides $n+2$.

2010 Pan African, 1
Tags: prime number , number theory
a) Show that it is possible to pair off the numbers $1,2,3,\ldots ,10$ so that the sums of each of the five pairs are five different prime numbers. b) Is it possible to pair off the numbers $1,2,3,\ldots ,20$ so that the sums of each of the ten pairs are ten different prime numbers?

2022 Germany Team Selection Test, 2
Tags: number theory , combinatorics , construction
Let $r>1$ be a rational number. Alice plays a solitaire game on a number line. Initially there is a red bead at $0$ and a blue bead at $1$. In a move, Alice chooses one of the beads and an integer $k \in \mathbb{Z}$. If the chosen bead is at $x$, and the other bead is at $y$, then the bead at $x$ is moved to the point $x'$ satisfying $x'-y=r^k(x-y)$. Find all $r$ for which Alice can move the red bead to $1$ in at most $2021$ moves.

2011 Albania National Olympiad, 2
Tags: euler , modular arithmetic , number theory
Find all the values that can take the last digit of a "perfect" even number. (The natural number $n$ is called "perfect" if the sum of all its natural divisors is equal twice the number itself.For example: the number $6$ is perfect ,because $1+2+3+6=2\cdot6$).

2004 USAMO, 2
Tags: vector , induction , integration , number theory , greatest common divisor , algorithm , relatively prime
Suppose $a_1, \dots, a_n$ are integers whose greatest common divisor is 1. Let $S$ be a set of integers with the following properties: (a) For $i=1, \dots, n$, $a_i \in S$. (b) For $i,j = 1, \dots, n$ (not necessarily distinct), $a_i - a_j \in S$. (c) For any integers $x,y \in S$, if $x+y \in S$, then $x-y \in S$. Prove that $S$ must be equal to the set of all integers.

2023 LMT Fall, 23
Tags: number theory
Let $S$ be the set of all positive integers $n$ such that the sum of all factors of $n$, including $1$ and $n$, is $120$. Compute the sum of all numbers in $S$. [i]Proposed by Evin Liang[/i]

2018 Israel National Olympiad, 4
Tags: number theory , digit , digit sum , divisibility
The three-digit number 999 has a special property: It is divisible by 27, and its digit sum is also divisible by 27. The four-digit number 5778 also has this property, as it is divisible by 27 and its digit sum is also divisible by 27. How many four-digit numbers have this property?

2016 Romania Team Selection Tests, 2
Tags: function , number theory , functional equation , algebra
Determine all $f:\mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ such that $f(m)\geq m$ and $f(m+n) \mid f(m)+f(n)$ for all $m,n\in \mathbb{Z}^+$

2018 India IMO Training Camp, 3
Tags: number theory , vieta jumping
Find the smallest positive integer $n$ or show no such $n$ exists, with the following property: there are infinitely many distinct $n$-tuples of positive rational numbers $(a_1, a_2, \ldots, a_n)$ such that both $$a_1+a_2+\dots +a_n \quad \text{and} \quad \frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_n}$$ are integers.

2006 Vietnam Team Selection Test, 2
Tags: number theory
Find all pair of integer numbers $(n,k)$ such that $n$ is not negative and $k$ is greater than $1$, and satisfying that the number: \[ A=17^{2006n}+4.17^{2n}+7.19^{5n} \] can be represented as the product of $k$ consecutive positive integers.